#
# ExB_Filter_Exercise_3.py
#
# This file is used to numerically integrate
# the second order linear differential equations
# that describe the trajectory of a charged particle through
# an E x B velocity filter.
#
# Here, it is assumed that the axis of the filter
# is aligned with the z-axis, that the magnetic field
# is along the +x-direction, and that the electric field
# is along the -y-direction.
#
# The numerical integration is done using the built-in
# routine odeint
#
# The specific goal of this code is to identify the
# maximumm value of v_z that permits transmission of the
# particles through the velocity filter with
# a specified exit aperture.
#
# By:
# Ernest R. Behringer
# Department of Physics and Astronomy
# Eastern Michigan University
# Ypsilanti, MI 48197
# (734) 487-8799 (Office)
# ebehringe@emich.edu
#
# Last updated:
#
# 20160624 ERB
#
from pylab import figure,plot,xlim,xlabel,ylim,ylabel,grid,title,show,legend
from numpy import sqrt,array,arange,linspace,zeros,absolute
from scipy.integrate import odeint
#
# Initialize parameter values
#
q = 1.60e-19 # particle charge [C]
m = 7.0*1.67e-27 # particle mass [kg]
KE_eV = 100.0 # particle kinetic energy [eV]
Ex = 0.0 # Ex = electric field in the +x direction [N/C]
Ey = -105.0 # Ey = electric field in the +y direction [N/C]
Ez = 0.0 # Ez = electric field in the +z direction [N/C]
Bx = 0.002 # Bx = magnetic field in the +x direction [T]
By = 0.0 # By = magnetic field in the +x direction [T]
Bz = 0.0 # Bz = magnetic field in the +x direction [T]
R_mm = 1.0 # R = radius of the exit aperture [mm]
L = 0.25 # L = length of the crossed field region [mm]
u = [1.0,1.0,100.0]/sqrt(10002.0) # direction of the velocity vector
Ntraj = 1000 # number of trajectories
# Derived quantities
qoverm = q/m # charge to mass ratio [C/kg]
KE = KE_eV*1.602e-19 # particle kinetic energy [J]
R = 0.001*R_mm # radius of the exit aperture [m]
vmag = sqrt(2.0*KE/m) # particle velocity magnitude [m/s]
vzpass = -Ey/Bx # z-velocity for zero deflection [m/s]
# Set up the array of z-velocities to try
vz = vzpass + linspace(-0.25*vzpass,0.25*vzpass,Ntraj+1) # the set of initial z-velocities
particle_pass = zeros(Ntraj+1)
#
# Over what time interval do we integrate?
#
tmax = L/vzpass;
#
# Specify the time steps at which to report the numerical solution
#
t1 = 0.0 # initial time
t2 = tmax # final scaled time
N = 1000 # number of time steps
h = (t2-t1)/N # time step size
# The array of time values at which to store the solution
tpoints = arange(t1,t2,h)
#
# Here are the derivatives of position and velocity
def derivs(r,t):
# derivatives of position components
xp = r[1]
yp = r[3]
zp = r[5]
dx = xp
dy = yp
dz = zp
# derivatives of velocity components
ddx = qoverm*(Ex + yp*Bz - zp*By)
ddy = qoverm*(Ey + zp*Bx - xp*Bz)
ddz = qoverm*(Ez + xp*By - yp*Bx)
return array([dx,ddx,dy,ddy,dz,ddz],float)
# Specify initial conditions that don't change
x0 = 0.0 # initial x-coordinate of the charged particle [m]
dxdt0 = 0.0 # initial x-velocity of the charged particle [m/s]
y0 = 0.0 # initial y-coordinate of the charged particle [m]
dydt0 = 0.0 # initial y-velocity of the charged particle [m/s]
z0 = 0.0 # initial z-coordinate of the charged particle [m]
# Start the loop over the initial velocities
for i in range (0,Ntraj):
# Specify initial conditions
dzdt0 = vz[i] # initial z-velocity of the charged particle [m/s]
r0 = array([x0,dxdt0,y0,dydt0,z0,dzdt0],float)
# Calculate the numerical solution using odeint
r = odeint(derivs,r0,tpoints)
# Extract the 1D matrices of position values
position_x = r[:,0]
position_y = r[:,2]
# Check if the particle made it through the aperture
if absolute(position_x[N-1]) < R:
if absolute(position_y[N-1]) < sqrt(R*R - position_x[N-1]*position_x[N-1]):
particle_pass[i] = 1.0
else:
particle_pass[i] = 0.0
else:
particle_pass[i] = 0.0
# Look for the specific value of
for i in range (int(Ntraj/2),int(Ntraj)):#Frem:Added int
if absolute(particle_pass[i]-particle_pass[i-1]) > 0.5:
print("i = %d"%(i-1),"vz[i] = %.3e"%vz[i-1]," m/s.")#Frem:Added brackets
Deltav = vz[i-1] - vzpass
print("Delta v = %.3e"%Deltav," m/s.")#Frem:Added brackets
Deltavovervzpass = Deltav/vzpass
print("Delta v/vzpass = %.3e"%Deltavovervzpass)#Frem:Added brackets
# start a new figure
figure()
# Plot the particle pass function versus z-velocity
plot(vz,particle_pass,"b-",label='$R = $%.2f mm'%R_mm)
xlim(min(vz),max(vz))
ylim(0.0,1.2)
xlabel("$v_z$ [m/s]",fontsize=16)
ylabel("Transmitted truth value",fontsize=16)
grid(True)
title('Wien filter: $v = $%.2e m, length $L = $%.2f m'%(vmag,L))
legend(loc=1)
show()
# start a new figure
figure()
# Plot the particle pass function versus scaled z-velocity
plot(vz/vzpass,particle_pass,"b-",label='$R = $%.2f mm'%R_mm)
xlim(min(vz/vzpass),max(vz/vzpass))
ylim(0.0,1.2)
xlabel("$v_z/v_{z,pass}$",fontsize=16)
ylabel("Transmitted truth value",fontsize=16)
grid(True)
legend(loc=1)
title('Wien filter: $v_{z,pass} = $%.2e m, length $L = $%.2f m'%(vzpass,L))
show()